We then think of P as a family of polynomials parametrized by the affine space As and the reduction is a “specialization of the parameters”

Reduction and specialization of polynomials

by

Pierre D`ebes (Lille)

1. Presentation. The main topic is the irreducibility of polynomials obtained by reduction or specialization from a polynomial P ∈ A[T, Y] with coefficients in an integral domain A and assumed to be irreducible over the algebraic closure k of the fraction field k of A. By “reduction” we mean reduction of the coefficients modulo some prime ideal p ⊂ A, and

“specialization” means specialization of the indeterminate T to some value t∈A (¹). The case that A is the ring of integers of some number field kis a typical situation. Another one is whenA=R[X1, . . . , Xs] is a polynomial ring insnew indeterminatesX1, . . . , Xs over some coefficient ring R, andp is the ideal generated byX₁−x₁, . . . , X_s−x_swithx₁, . . . , x_s∈R. We then think of P as a family of polynomials parametrized by the affine space A^s and the reduction is a “specialization of the parameters”.

The Bertini–Noether theorem for reduction—P modulo p is irreducible over the algebraic closure κ_p of the fraction field κ_p of A/p, for all primesp but in a proper closed Zariski subset of Spec(A)—and Hilbert’s irreducibility theorem (HIT) for specialization—ifAis the ring of integers of some number fieldk, thenP(t, Y) is irreducible ink[Y] for infinitely manyt∈A—are the fundamental results. Among existing methods for these two results, some have a common trend which is to reduce modulo “good primes”. This mainly refers to the Grothendieck good reduction theorem (GRT) and the so-called congruence approach to HIT, notably developed by Eichler [Eic39], Fried [Fri74], Fried–Jarden [FJ04, §13.3], Ekedahl [Eke90], Colliot-Th´el`ene and Serre [Ser92, §3.5].

2010Mathematics Subject Classification: Primary 12E05, 12E25, 14E20, 12Yxx; Secondary 12E30, 12Fxx, 14Gxx.

Key words and phrases: polynomials, reduction, specialization, Bertini–Noether theorem, Hilbert irreducibility theorem, Grothendieck good reduction criterion.

Received 26 May 2015; revised 18 September 2015.

Published online *.

(¹) Specialization is also a reduction of P, but viewed as a polynomial in Y with coefficients inA[T]; to avoid confusion we prefer to use two different words.

DOI: 10.4064/aa8176-12-2015 [1] c Instytut Matematyczny PAN, 2015

We contribute to this trend. Our approach rests on the same funda- mental results but introduces a certain tool that somewhat unifies both the reduction and specialization questions. We obtain fully explicit statements, phrased in terms of polynomials rather than the corresponding algebraic cov- ers (²) and which have new applications to issues around the effective Hilbert irreducibility theorem. The following statement gives an idea of our results.

(Theorem3.1) Assumek is a number field. There exist integersN, B, C and a finite extension L/Q such that the following holds. If p₁, . . . , p_N areN distinct prime numbers satisfying

(∗) p_i-B, p_i ≥C andp_i is totally split inL/Q, i= 1, . . . , N,

then for any multiple a ∈ Z of p1· · ·pN, there exists b ∈ N such that the polynomialP(am+b, Y) is irreducible in k[Y]for every integer m.

N,B,C,Lare precisely given in§3, making Theorem 3.1 totally explicit and improving on previous results which only showed the existence of some cosetsaZ+bsatisfying the conclusion.

The main ingredients are the GRT and the last variant of the congruence approach, developed in [DG12], [DL13], [DL12], which we adjust and recast in our context; Theorem 3.1 is an explicit polynomial version of [DL12, Corollary 4.5].

Our new tool is the bad prime divisor of P. It is a certain non-zero parameterB_P ∈A, which is directly computable from the coefficients ofP through elementary operations, starting with the discriminant ∆P ∈ A[T]

of P relative to Y (see §2), and which articulates both the reduction and specialization issues. The integerB of Theorem 3.1 can be taken to be the normN_k/Q(B_P).

The point ofB_P is this. WhenAis a Dedekind domain of characteristic 0, the non-zero prime idealsp⊂AdividingB_P are those for which some of the distinct roots of∆_P become equal or infinite modulop. Such a prime ideal is called bad and the other ones are thegood primes. Our bad/good primes relate to more geometric versions previously introduced; an advantage of B_P is thatit behaves well under morphisms: if p is a good prime such that deg_Y(P)!∈/p, we have

(Theorem2.6-a) B_Pmodp=B_P modp and is non-zero in A/p.

This is one conclusion of Theorem 2.6, which contains the gist of the whole reduction-specialization approach. Forpas above, we have the Bertini–

Noether conclusion:

(²) The difference is that the notion of cover identifies all the polynomial equations of a given cover. This is illustrated by the example ofP =Y²−4T which, depending on whether it is viewed as a cover or a polynomial, has good reduction at 2 or not.

(2.6-b) The polynomial P modulop is irreducible in κp[T, Y] (³).

Furthermore, whenAis the ring of integers of a number fieldk, we have the following “Chebotarev” conclusion (⁴): if in additionpis of suitably large norm (a condition leading to the integerC of Theorem 3.1) and satisfies one last assumption (below), then

(2.6-c) Each element of the Galois groupG of P over k(T)is the Frobe- nius at p of the splitting field of some specialization P(t0, Y) (t0∈A).

The last assumption onp involves another important phenomenon: the possible appearance of new constants in the splitting field of P overk(T), and the assumption is that p be totally split in the field of new constants (the field L of Theorem 3.1). For example, for P = Yⁿ −T ∈ Q[T, Y], nth roots of unity appear. Proposition 2.8 collects some information on this phenomenon, some of it new to our knowledge. But it remains mysterious and hard to control. In “most cases” however, no constant extension occurs and so the condition on p disappears. By “most cases” we mean that this holds if G=Sn (with n= deg_Y(P)), which in various senses is the generic case.

We postpone the proof of Theorem 2.6 to§5 and prefer to show first how this statement leads to explicit versions of Hilbert’s irreducibility theorem, for one polynomial in §3, and for a family of polynomials in §4.

Theorem 3.1 discussed above is a first application. Fork=Q, we deduce bounds for the least integer t ≥0 such thatP(t, Y) is irreducible in Q[Y], along with some algorithms to find it. Bounds known so far [D`eb96], [SZ95], [Wal05], [DW08], involve these parameters:

deg_Y(P) =n, deg_T(P) =m and H(P) = height ofP.

Finding a bound not depending on H(P) and so depending only on the degree of P is an important problem. It is in particular a non-trivial test for the far-reaching Lang conjecture on the rational points on an algebraic variety of general type over a number field; this conjecture is indeed known to imply that such a bound exists [DW08,§5]. Our bounds only involve the degree nand the discriminant ∆_P:

(Corollaries 3.6 & 3.8) The bounds we obtain depend on n= deg_Y(P), ρ=number of distinct roots of ∆P

(³) A proof of this, not using the GRT as we do but in the special case that the residue characteristic is positive, can be found in [Zan97].

(⁴) This conclusion already appears in some form in earlier papers without the explicit conditions on p [FJ04, Lemma 13.3.4], [Eke90, Lemma 1.2]. Our explicitness precisely follows from the use of the GRT which replaces the ineffective use of the Bertini–Noether theorem.

and







the numberβ of bad primes of P in “most cases”, the bad primes ofP and the prime factors

of some non-zero integral value ∆_P(τ) in all cases (⁵).

Our bounds are larger than those of [Wal05] which are more appropriate for an algorithmic use. The interest of ours lies in the parameters they depend on. First, the dependence onn, ρis better than that onn,m: indeed, ρ ≤ deg(∆_P) ≤ (2n−1)m. Furthermore, in the generic case G = S_n, the parametersn,ρ,β are numerical degrees which do not involve the height of the coefficients of P. Finally, the bounds we obtain whenn, ρ, β are fixed still concern infinitely many polynomials, while there are only finitely many P ∈Z[T, Y] withn,m,H(P) bounded: for example forP_a=Yⁿ−aY −T (a∈Z), for which G =Sn, we have ρ =n−1 and the bad primes are the prime divisors ofn(n−1)a.

In §4, in the context of a family of polynomials—P ∈ A[T, Y] with A=R[X₁, . . . , X_s] andR the ring of integers of some number field k—our goal is to investigate to what extent the bounds given by§3 for each individ- ual polynomial P(x1, . . . , xs, T, Y) ∈ R[T, Y] depend on (x1, . . . , xs) ∈ R^s. An ideal conclusion would be to have a constant global bound. Using non- standard analysis, Yasumoto could reach this goal for s = 1 [Yas87]; he mentioned that the case s > 1 seemed very difficult. In the general situ- ation s ≥ 1, our approach leads to a “family of bounds”. In the generic case G = S_n, these bounds involve polynomials in (x₁, . . . , x_s) with coeffi- cients inR. Some of these polynomials can be made constant, for example deg(P(x1, . . . , xs, T, Y)) ≤ deg_{(T ,Y)}(P), but there remains one that is not, which is the bad prime divisor. Our result shows some progress for s > 1 but the ideal goal remains to be attained.

2. Bad prime divisor. Let A be an integrally closed domain with fraction field k.

2.1. Reduced polynomial. Thereducedform∆^redof a non-zero poly- nomial ∆ ∈A[T] is defined as follows. Denote the leading coefficient of ∆ by∆0 and its distinct roots byt1, . . . , tρ∈k. Then

∆^red=∆^ρ₀ Y

1≤i≤ρ

(T −t_i) and∆^red is defined to be 1 if ∆∈A (i.e. ρ= 0).

(⁵) There is an intermediate situation: if the genus of the splitting field of P is at least 2, the bound depends only onn,ρand the bad primes ofP (Corollary 3.7).

Consider the factorization∆=∆0Q`

j=1Q^α_j^j of∆ink[T] withQ1, . . . , Q_` the distinct irreducible and monic factors ink[T]. We will use several times the following condition on ∆:

(∗) The polynomialsQ₁, . . . , Q_` are separable, i.e. have no multiple roots in k.

This holds in particular if k is of characteristic 0 or greater than deg(∆).

Lemma 2.1. Assume that the polynomial ∆ satisfies(∗). Then

∆^red=∆^ρ₀ Y

1≤j≤`

Q_j =∆^ρ−1₀ ∆ gcd(∆, ∆⁰)

where the gcd is calculated in the ring k[T]and made monic by multiplying by a suitable non-zero constant. Furthermore, ∆^red is a polynomial with coefficients in A.

Proof. The two polynomials in the first equality have only simple roots (by (∗)) and have the same sets of roots. The equality follows since they also have the same leading term. The second equality rests on the fact that if

∆=∆0Qρ

j=1(T−t_j)^βj withβ1, . . . , βρ≥1, then, up to some non-zero factor in k, gcd(∆, ∆⁰) = Qρ

j=1(T −tj)^βj−1. Finally, ∆0t1, . . . , ∆0tρ are integral over A. Hence so are the coefficients of ∆^red = Q

1≤i≤ρ(∆₀T −∆₀t_i). As they are also ink, and A is integrally closed, they are inA.

Remark 2.2. The second formula, which makes it possible to calculate

∆^red thanks to the Euclidean algorithm, is useful in practice.

2.2. Reduced discriminant and bad prime divisor. LetP∈A[T, Y] be a polynomial such that deg_Y(P) =n≥1 and assumed to be monic and separable inY.

Remark2.3. A standard transformation makes it possible to reduce to the situation where P is monic in Y. Namely replace

P(T, Y) =P0Yⁿ+P1Yⁿ⁻¹+· · ·+Pn

withP₀, P₁, . . . , P_n∈A[T] by

Q(T, Y) =P₀ⁿ⁻¹P(T, Y /P0) =Yⁿ+P1Yⁿ⁻¹+· · ·+P₀ⁿ⁻¹Pn. It is convenient, for the theory and in practice, to start with this transfor- mation when studying the reduction and reducibility properties of a poly- nomialP. The factorP₀ is retrieved and so is implicitly kept track of in the discriminant of the polynomial Q(T, Y).

Denote thediscriminant ofP relative toY by

∆P = discY(P).

It is a non-zero polynomial inA[T] of degree≤(2n−1) deg_T(P). Consider thereduced discriminant

∆^red_P = (∆P,0)^ρ

ρ

Y

i=1

(T −ti)∈A[T]

where ∆_P,0 is the leading coefficient of ∆_P and t₁, . . . , t_ρ are the distinct roots of ∆_P in k. Assume further that ∆_P satisfies condition (∗). Then

∆^red_P ∈A[T] and its discriminant

disc(∆^red_P ) = (∆_P,0)^2ρ(ρ−1) Y

1≤i6=j≤ρ

(t_j−t_i)

is an element ofA, non-zero since by construction∆^red_P has no multiple root ink(⁶). Define then an elementB_P by

B_P =∆_P,0·disc(∆^red_P ).

We have B_P ∈A andB_P 6= 0.

Definition 2.4. The maximal ideals p⊂A that containB_P are called the bad primes of P ∈ A[T, Y], and B_P is called the bad prime divisor.

Maximal idealsp⊂A that are not bad are said to begood.

Remark 2.5. (a) IfA₀ ⊂A is an integrally closed subring and P is in A₀[T, Y], the bad prime divisors relative toA₀ and to Acoincide.

(b) The resultant

res(∆^red_P ,(∆^red_P )⁰)

is an alternative definition of B_P. It is indeed equal to the discriminant disc(∆^red_P ) multiplied by the leading term∆^ρ_P,0 of∆^red_P , and so the set of bad primes remains the same. Similarly the polynomial res(P, P⁰) can be used instead of ∆_P at the beginning of the construction. In practice, it can be advantageous to use the resultant rather than the discriminant; the former is indeed easier to compute from its Sylvester definition as a determinant.

In the same vein, one may replace ∆^ρ_P,0 by ∆^ρ_P,0⁰ with some ρ⁰ ≥ ρ (e.g.

ρ⁰ = deg(∆P)) in the definition of ∆^red_P ; this will not affect the set of prime divisors ofB_P.

2.3. The central result. Retain the notation introduced in §2 and assume further thatAis a Dedekind domain.

Let G be the Galois group of the splitting field of P over k(T); G is called the monodromy group of P, it is a transitive subgroup of S_n with n= deg_Y(P). Also denote by bk_P/k the constant extension in the splitting fieldF /k(Tb ) of the polynomial P, i.e. bkP =Fb∩k.

(⁶) When∆P ∈A, e.g. when deg_T(P) = 0, then∆^redP = 1 and disc(∆^redP ) = 1.

If p ⊂ A is a non-zero prime ideal, denote the residue field A/p by κp, the reduction map by sp :A →κp, the localized ring of A by p by Ap, and the polynomial obtained by reducing the coefficients of P by s_p(P).

Theorem2.6. Assumekis of characteristic0or greater thandeg(∆P).

Let p⊂A be a good prime of P such that|G|∈/ p. Then we have these three conclusions:

(Good Behaviour) We have

∆^red_s

p(P)=s_p(∆^red_P )6= 0, B_sp(P)=s_p(B_P)6= 0.

(GRT) If P is irreducible in k(T)[Y], the polynomial s_p(P) is monic, separable in Y, irreducible in κp[T, Y] and of monodromy groupG.

Assume further that k is a number field and A is its ring of integers.

(Chebotarev) If p is totally split in the extension bkP/k and of norm N_k/Q(p) ≥ (ρ+ 1)²|G|², then for every ω ∈ G, there is an element t_p ∈ A such that∆_P(t_p)∈/ pand for everyt∈A_pwitht≡t_pmodulopthe Frobenius at p of the splitting field ofP(t, Y) over bkP is conjugate to ω in G.

The condition B_sp(P) = sp(B_P) 6= 0 can be equivalently rephrased as saying that no distinct roots ti and tj of ∆P coincide modulo p, and none of the roots t_i is∞ modulo p.

Theorem 2.6 and its Addendum 2.8 below are proved in §5. The GRT part is essentially the Grothendieck good reduction criterion. The Cheb- otarev part is deduced from revisiting the results of [DG12] and making them explicit. The Good Behaviour part is new.

2.4. First two examples. We give two examples illustrating Theo- rem 2.6.

2.4.1.SpecializationsP(t, Y)with no root. The first one is an immediate consequence of the Chebotarev part. Assumek=Qfor simplicity.

Corollary2.7. If p=pZis as in(Chebotarev), there is a cosetpZ+b

⊂Z such that for each t∈pZ+b, P(t, Y) has no roots in Q.

Proof. The subgroup G ⊂ Sn being transitive, there classically exists ω ∈ G with no fixed points. The result follows from the Chebotarev conclu- sion applied to this ω.

2.4.2.Successive specializations. LetP ∈k[X₁, . . . , X_s][T, Y] be a poly- nomial, monic, separable in Y and assumed to be irreducible in the ring k(X1, . . . , Xs)[T, Y]. Consider its bad prime divisor B_P ∈ k[X1, . . . , Xs].

Theorem 2.6 can be applied to P viewed as being in k(X1, . . . , Xs−1)[Xs]

[T, Y]. The bad prime divisor of P relative to k(X1, . . . , Xs−1)[Xs] is the same as relative to the smaller ringk[X1, . . . , Xs] (Remark 2.5). Hence it is the polynomialB_P ink[X₁, . . . , X_s] introduced above.

Assume kis of characteristic 0 for simplicity; the condition |G|∈/ pthen always holds. Letx_s∈kbe such thatB_P(X₁, . . . , Xs−1, x_s)6= 0. From Theo- rem 2.6,P(X1, . . . , Xs−1, xs, T, Y) is monic, separable inY and irreducible in k(X1, . . . , Xs−1)[T, Y], and its bad prime divisor isB_P(X1, . . . , Xs−1, xs)∈ k[X₁, . . . , Xs−1]. Theorem 2.6 can then be applied to P(X₁, . . . , Xs−1, x_s, T, Y) to specialize Xs−1. An inductive argument finally leads to this con- clusion:

(∗) If (x1, . . . , xs) ∈k^s satisfies B_P(x1, . . . , xs) 6= 0 in k, then the poly- nomial P(x₁, . . . , x_s, T, Y)∈k[T, Y]is irreducible in k[T, Y].

A variant of this argument will be used in §4.4.

2.5. More on the extensionbkP/k. LetF/k(T) be the extension gen- erated by some root ofP (as a polynomial inY). Recall that the constant extensionin F/k(T) is the extension F ∩k/k. If the polynomial P is irre- ducible ink[T, Y], we haveF∩k=k; the extensionF/k(T) is then said to beregular over k. The extensionbkP/k is the constant extension in the Ga- lois closure F /k(Tb ) of F/k(T). It need not be trivial even though F/k(T) is regular over k. In general, if G_a = Gal(F /k(T)), the extensionb bkP/k is Galois of groupG_a/G.

Addendum 2.8 (on the constant extension).

(a) We havebk_P =k in each of the following situations:

(a-1) P is irreducible ink(T)[Y]and the extensionF/k(T)is Galois and regular over k,

(a-2) G=Sn.

(b) (Complement to (GRT)) If P is irreducible in k(T)[Y], p a good prime of P and |G| ∈/ p, then the residue extension of bk_P/k at some/any prime abovepcontains the constant extension in the split- ting field ofs_p(P) over κ_p(T).

(c) If k is a number field and Fb is a function field of genus ≥ 2, then bk_P/k is one from the finite list of extensions of k of degree

≤ |Nor_Sn(G)|/|G|and unramified at each good prime of P.

In §3 and §4, we use Theorem 2.6 and its Addendum 2.8 to deduce quite precise versions of HIT for a single polynomial and for a family of polynomials over number fields.

3. Hilbert irreducibility theorem. We retain the notation intro- duced in previous sections and assume further that k is a number field, A is its ring of integers andP ∈A[T, Y] is irreducible ink[T, Y].

3.1. From Chebotarev to Hilbert. A standard argument easily con- nects the Chebotarev conclusion from Theorem 2.6 to a Hilbert conclusion.

Namely if C1, . . . , CN are N conjugacy classes of G and p₁, . . . ,p_N are N non-zero prime ideals of A that are good, of norm N_k/Q(p) ≥(ρ+ 1)²|G|² and totally split in the extension bk_P/k, then the Chebotarev conclusion combined with the Chinese remainder theorem provides an element b∈ A with the following property:

(∗) For every t ∈ Ap1 ∩ · · · ∩ ApN, in particular for every t ∈ A, if t ≡ bmodp₁· · ·p_N, then the Galois group over bk_P of the poly- nomial P(t, Y) contains elements of each of the conjugacy classes C₁, . . . , C_N.

Furthermore, under the assumption that the respective prime numbers p₁, . . . , p_n ∈ Z below p₁, . . . ,p_N are totally split in the extension bk_P/Q (and not just p₁, . . . ,p_N inbkP/k), then bcan be chosen in Z.

Hence ifC1, . . . , CN are initially chosen so that

(∗∗) for any(g1, . . . , gN)∈C1× · · · ×CN, the subgrouphg₁, . . . , gNi ⊂ G acts transitively on {1, . . . , n},

then the Galois group of P(t, Y) acts transitively on {1, . . . , n}; conse- quently, P(t, Y) is irreducible in bk_P[Y] anda fortiori ink[Y].

Such a choice of C₁, . . . , C_N is always possible: from a classical lemma of Jordan [Jor72], (∗∗) holds ifC1, . . . , C_N are all the non-trivial conjugacy classes of G; and then even more holds since in this case we have in fact hg₁, . . . , g_Ni =G. Therefore if NG is the smallest integer N such that (∗∗) holds and cc(G) is the number of conjugacy classes of G, then

NG<cc(G)≤ |G| ≤n!.

However NG can be smaller than these bounds: for example, if G contains an n-cycle, thenNG= 1.

3.2. Main result. We obtain the following version of Hilbert’s irre- ducibility theorem.

Theorem 3.1. For the integers N, B, C and the finite extension L/Q specified below, we have the following. If p1, . . . , p_N are N distinct prime numbers satisfying

(∗) pi-B, pi≥C and pi is totally split in L/Q, i= 1, . . . , N,

then for any multiplea∈Zof p1· · ·pN, there existsb∈Nsuch thatP(am+

b, Y) is irreducible in k[Y] for every integerm.

Addendum 3.2 (on the constants). One can take (a) N =NG (see§3.1),

(b) B=N_k/Q(B_P) (norm of the bad prime divisor),

(c) C= (ρ+1)²|G|² (ρis the number of distinct roots of the discriminant

∆P andG is the monodromy group ofP), (d) L=bk_P (constant extension in Galois closure).

Remark3.3 (on the constants). (a) One can also take forBthe product of all distinct primesp∈N dividingN_k/Q(B_P).

(b) Recall some estimates which relate our parameters to other classical ones; here D = deg(P), r is the branch point number of the extension F/k(T) associated withP(T, Y) andg its genus, i.e. the genus of the curve P(t, y) = 0:







r≤ρ+ 1≤deg(∆_P) + 1≤(2n−1)m+ 1 (ρ= deg(∆^red_P );§2) r/2 + 1−n≤g≤rn/2 + 1−n−r/2 (Riemann–Hurwitz) g≤ ¹₂(D−1)(D−2) [FJ04, Corollary 5.3.5].

Remark3.4 (on the statement). (a) Denote byH_P the Hilbert subset of allt∈ksuch thatP(t, Y) is irreducible ink[Y]. The conclusion of Theorem 3.1 can be rephrased as saying that for any multiple a ∈ Z of p₁· · ·p_N, the Hilbert subsetH_P contains at least one coset moduloa, or equivalently that, for some b ∈ N, the Hilbert subset associated with the polynomial P(aT +b, Y) contains all integers.

(b) The statement readily extends to several polynomials P1, . . . , P` ∈ A[T, Y]: if N<sub>j</sub>, B<sub>j</sub>, C<sub>j</sub>, L<sub>j</sub> are given by Theorem 3.1 for the polynomial P<sub>j</sub>, then taking N = N<sub>1</sub> +· · ·+N<sub>`</sub>, B = B₁· · ·B_{`</sub>, C = max(C<sub>1</sub>, . . . , C<sub>`}) and L/Q the compositum of the extensions L1/Q, . . . , L_{`</sub>/Q and using the Chinese remainder theorem yields the conclusion of Theorem 3.1 with H<sub>P1</sub>∩ · · · ∩ H<sub>P`}∩Z replacingH_P ∩Z.

(c) Using (b) one can relax the assumption thatP(T, Y) is irreducible in Q[T, Y] to only assume that it is irreducible ink[T, Y]. A classical reduction [D`eb09, lemme 5.1.3] indeed shows that

H_P ⊃ H_P1∩ · · · ∩ H_P`(up to some finite set (⁷))

with P1, . . . , P_` the irreducible factors of P in Q[T, Y]. This reduction in- volves a finite extension of the base field k and so requires to apply Theo- rem 3.1 to that bigger number field.

(⁷) Of cardinality depending only on deg(P).

(d) Combining (b) and (c) shows that Theorem 3.1 extends to the situ- ation where H_P is replaced by a general Hilbert subset of k, i.e. the inter- section of several subsets H_P withP irreducible ink[T, Y].

(e) Proceeding as in §3.1 but with a bigger set {p₁, . . . ,p_N,q₁, . . . ,q_M} of primes (satisfying the assumptions of the Chebotarev conclusion of The- orem 2.6), one obtains an improved conclusion of Theorem 3.1 for which in addition to P(am+b, Y) being irreducible in k[Y], the Frobenius of the splitting field ofP(am+b, Y) overbk_P at each primeq_j can be prescribed to be conjugate to an arbitrarily given element of G,j = 1, . . . , M. This yields the type of conclusions that are given in previous papers in a geometric context; see for example [DL12, Corollary 4.5].

3.3. Bounds for the least specialization in H_P. Assume k = Q. Denote byβthe number of bad primes ofP, i.e. the number of prime factors ofB_P. The general idea is to minimizeNGand evaluate the minimum integer M such that the interval [(ρ+ 1)²|G|², M] containsβ+NG primes that are totally split in QbP/Q. This interval then automatically contains NG primes satisfying condition (∗) from Theorem 3.1, which then produces a coset aZ+b⊂ H_P with athe product ofNG such primes.

In§3.3.1 and§3.3.2, we focus on the search of the integera. This leads to bounds for the least positive integer inH_P. In §3.3.3, we make some further algorithmic comments on the search of b.

3.3.1. Special case QbP =Q. In this case the condition that the primes are totally split inQbP/Qdisappears and we obtain this statement.

Corollary 3.5. If QbP = Q, e.g. in each of the situations (a-1) and (a-2) from Addendum 2.8, the Hilbert subset H_P contains a coset aZ+b with aand b smaller than some bound depending only on n, ρ and β.

In particular, whenG=Sn, we have QbP =Qand NG = 1.

Corollary3.6. Assumek=QandG=S_n. Then for every good prime number p ≥((ρ+ 1)n!)², there is a coset pZ+b ⊂Z such that P(t, Y) is irreducible inQ[Y]for every t∈pZ+b. Furthermore, p and bcan be taken as follows and can be bounded in terms of n, ρ, β:

• p: the first good prime of P that is ≥((ρ+ 1)n!)²;

• b: any integer such that P(b, Y) modulo p is irreducible in Fp[Y].

Proof. Here the general strategy applies with NG = 1 and C1 the con- jugacy class of n-cycles. For p chosen as indicated, the method guarantees the existence of integers b such that ∆_P(b) ∈/ pZ and the Frobenius of the splitting field ofP(b, Y) atpis ann-cycle. Due to the condition∆_P(b)∈/ pZ, this is equivalent toP(b, Y) modulopbeing irreducible inFp[Y], whence the result.

3.3.2.General caseQbP ⊃Q. In this case, explicit information is needed on the primes that are totally split in the extensionQbP/Q. For this, one can use effective versions of the Chebotarev density theorem [LO77], [LMO79], [Ser81]. These results involve the order of the Galois group Gal(QbP/Q) and the discriminant|d

QbP|, and so lead to bounds depending on n,ρ and |d

QbP|.

The parameter|d

QbP|may however be hard to control. We offer an alternative approach leading to the following results.

Corollary 3.7. Assume that k = Q and Fb is the function field of a curve of genus gb≥2. Then the Hilbert subset H_P contains a coset aZ+b with a and b smaller than some bound depending only on n, ρ and the set of bad primes of P.

Proof. From Addendum 2.8, QbP/Q is one from the finite list of exten- sions ofQof degree≤ |Nor_Sn(G)|/|G|and possibly ramified only at the bad primes of P. Therefore the firstNG primes satisfying condition (∗), and so the integersa and b from Theorem 3.1, can be bounded in terms n,ρ and the set of bad primes ofP.

When 0 ≤bg ≤1, one can reduce to the situation bg ≥2, at the cost of losing the conclusion thatH_P contains a whole coset.

Corollary 3.8. Assume k = Q. Let τ ∈ Z be such that ∆_P(τ) 6= 0.

Then the Hilbert subsetH_P contains a positive integert₀ smaller than some bound depending on n, ρ, the set of bad primes of P and the set of prime factors of∆_P(τ).

Proof. We adjust a reduction argument given in [DW08, §5.1]. More specifically, it follows from [D`eb92, Lemma 0.1] and ∆_P(τ) 6= 0 that for every integerh≥1, the polynomial

P_h(T, Y) =P(T^h+τ, Y)

is irreducible in Q[T, Y]. The Riemann–Hurwitz formula then shows that the genusg_h of the curve with affine equationP(t^h+τ, y) = 0 satisfies

2g_h−2≥ −2n+h.

Thus forh= 2n+ 2, we obtaing_h ≥2, and so is the genusbg_h of the splitting field ofP_h sincebg_h ≥g_h. From Corollary 3.7, there exists b∈Zless than a bound depending only on deg_Y(Ph), deg(∆^red_P

h) and the set of bad primes of P_h such that

Ph(b, Y) =P(b^h+τ, Y)

is irreducible in Q[Y]. Clearly deg_Y(Ph) = deg_Y(P) = n, and it is easily checked that

∆^red_Ph(T) =∆^red_P (T^h+τ).

From Remark 2.5, disc(∆^red_P

h) has the same set of prime divisors as the resultant of ∆^red_P

h and of (∆^red_P

h)⁰. We have res(∆^red_P

h,(∆^red_P

h)⁰) = (res(∆^red_P ,(∆^red_P )⁰))^h·∆^red_P (τ).

We conclude that the bad primes of P_h consist of the bad primes of P and of the prime factors of∆^red_P (τ).

3.3.3. Algorithmic remarks to find b. Assume that NG good primes p₁, . . . , p_NG ≥ (ρ+ 1)²|G|², totally split in QbP/Q, have been found. From Theorem 3.1, forathe product of these primes, the Hilbert subsetH_P con- tains a coset aZ+b for some integerb∈[1, a].

(a) An obvious option to find an element b ∈ H_P is to test all polyno- mials P(t, Y), t = 1, . . . , a; there are efficient irreducibility algorithms for polynomials in one indeterminate. At least one of these polynomialsP(t, Y) is irreducible in Q[Y].

(b) WhenG =Sn, Corollary 3.6 provides a simple algorithm to find an element ofH_P. This algorithm can be applied in the general context as a first step: forpchosen as indicated, ifP(b, Y) modulopis irreducible inFp[Y] for someb∈ {1, . . . , p}, thena fortiori,P(b, Y) is irreducible inQ[Y]; and if no polynomial P(b, Y) modulo p is irreducible in Fp[Y] (b ∈ {1, . . . , p}), then one should conclude that G 6=S_n and apply the method in a more refined way.

(c) In Corollary 3.6, the assumption thatG contains an n-cycle made it simple to determine the integer b; it can be generalized to assume that

(∗∗) the group G contains N elements ω1, . . . , ωN with the property that any N elements g₁, . . . , g_N ∈ S_n conjugate to ω₁, . . . , ω_N in S_n (respectively) generate a transitive subgroup of S_n.

Assumption (∗∗) holds ifG contains ann-cycle and in other situations.

For example, the groupGbeing transitive, it contains an element σwith no fixed points. Hence if G also contains an (n−1)-cycle (or more generally the product of anm-cycle and an (n−m)-cycle (0≤m≤n) with supports non-stable underσ), then condition (∗∗) holds. However (∗∗) does not always hold: for example it does not if G ⊂Sn is the regular representation and G is a non-cyclic p-group.

Under assumption (∗∗), pick N good primes p₁, . . . , p_N ≥ (ρ+ 1)²|G|², totally split in QbP/Q, and b1, . . . , b_N ∈ Z such that∆_P(bi) ∈/ piZ and the factorization type of P(bi, Y) modulopi is in the conjugacy class Ci of ωi, i = 1, . . . , N. The Frobenius g_i ∈ G of the splitting field of P(b_i, Y) at p_i then lies in C_i, i = 1, . . . , N. From (∗∗), g₁, . . . , g_N generate a transitive subgroup ofSn. We conclude that ifbis an integer such thatb≡bi modpi, i= 1, . . . , N, thenaZ+b⊂ H_P.

4. Families of polynomials. As in§3,kis a number field. Denote its ring of integers by R and consider the polynomial ring A =R[X1, . . . , Xs] insnew indeterminatesX₁, . . . , X_soverR. SetX=X₁, . . . , X_s to simplify the notation; soA=R[X].

Fix a polynomialF(X, T, Y)∈R[X, T, Y], irreducible ink(X)[T, Y] and monic in Y. Set deg_Y(F) =n and deg_T(F) =m, assume n≥1 and let G be the monodromy group of F, i.e. the Galois group of F(X, T, Y) over k(X)(T).

4.1. Qualitative statement of the main result. As for Theorem 3.1, we start with a qualitative statement and specify the parameters involved in a second stage.

Theorem 4.1. We produce below:

• a non-zero polynomial B_F ∈R[X],

• two integersN and C,

• a finite Galois extension L of k(X),

with the following property. For every x∈R^s such that B_F(x)6= 0, (a) the polynomialF(x, T, Y) is irreducible in k[T, Y],

(b) if p₁, . . . , p_N are distinct prime numbers satisfying

(∗) p_i-N_k/Q(B_F(x)), p_i ≥ C and p_i is totally split in the residue field Lx of L at X=x, i= 1, . . . , N,

anda is any multiple of p1· · ·pN, then there exists b∈Z such that F(x, t, Y) is irreducible in k[Y] for everyt in the coset aZ+b.

For every x∈k^s, set

P_x(T, Y) =F(x, T, Y).

From the Bertini–Noether theorem, for everyx∈R^sbut in a proper Zariski closed subsetE_F ⊂A^s(k), the polynomialF(x, T, Y) is irreducible ink[T, Y].

If x ∈R^s\ E_F, Theorem 3.1 can be applied to the polynomial Px(T, Y) ∈ R[T, Y].

Consider the integers N_x, B_x, C_x and the finite extension L_x/Q that are given by Theorem 3.1. Our goal is to investigate to what extent they depend on x. Recall that Nx, Bx, Cx and Lx can be bounded in terms of deg_Y(P_x) and further parameters involving the discriminant relative to Y and the bad prime divisor of P_x. Note right away that the first one can be bounded independently ofx: indeed, deg_Y(Px) = deg_Y(F) =n.

4.2. The Zariski closed subset E_F and the polynomial B_F. The- orem 2.6 explicitly provides a proper Zariski closed subset E_F. Denote by BF the bad prime divisor ofF(X, T, Y). It is a non-zero element of R[X].

Furthermore one can bound deg(B_F). Namely, it follows from (deg_T(∆F)≤(2n−1)m,

deg_X(∆F)≤(2n−1) deg_X(F) that

(deg_T(∆^red_F )≤(2n−1)m,

deg_X(∆^red_F )≤(2n−1)²mdeg_X(F).

The first inequality is obvious as deg_T(∆^red_F ) ≤deg_T(∆F). The second one follows from the inequality deg_X(∆^red_F ) ≤ deg_T(∆F) deg_X(∆F) (which is left as an exercise using Gauss’ lemma and∆^red_F ∈R[X][T]). The definition B_F =∆F,0·disc_T(∆^red_F ) then leads to

deg(B_F)≤(2n−1)²mdeg_X(F) + 2(2n−1)m−1

(2n−1)²mdeg_X(F)

≤16n³m²deg_X(F).

Denote the zero set of B_F in k^s by Z(B_F). Applying Theorem 2.6 to F(X, T, Y) ∈ A[T, Y] with A = k[X] and p = hX−xi ⊂ A with x ∈ k^s yields:

(∗) If x /∈ Z(B_F), then F(x, T, Y) = Px(T, Y) is irreducible in k[T, Y], of monodromy group G, and we have

∆^red_Px =∆^red_F (x), B_F(x) =B_Px 6= 0.

In particular, one can takeE_F =Z(B_F).

Fixx∈R^s\ Z(B_F). We study below each of N_x,B_x,C_x,L_x. 4.3. The integer N_x. From Addendum 3.2, one can take

N_x =NG.

In particular, N_x can be bounded independently of x.

4.4. The integer B_x. From Addendum 3.2, one can take B_x =N_k/Q(B_Px)

where B_Px is the bad prime divisor of P_x considered in R[T, Y]. From Re- mark 2.5, it is the same ifP_x is considered ink[T, Y]. Use then (∗) above to conclude that the integerBx can be taken to be

Bx =N_k/Q(B_F(x)),

which is the norm of the value at x of the polynomial B_F ∈ R[X] which depends only on the original polynomial F(X, T, Y).

4.5. The integer Cx. By construction, we have ∆^red_F ∈ R[X, T] and

∆^red_Px ∈ R[T], and from (∗) above, ∆^red_Px =∆^red_F (x). Denote by ρ the degree of these polynomials inT. From Addendum 3.2, one can take

C_x = (ρ+ 1)²|G|². In particularCx can be bounded independently ofx.

4.6. The extension L_x/Q. From Addendum 3.2, one can take L_x=bk_Px,

the constant extension in the splitting field ofPx overk(T).

Consider the constant extensionbkF/k(X) in the splitting field ofF over k(X, T). From Addendum 2.8, the field extension bkPx/k is contained in the residue extension, say (bkF)_x/k, of bkF/k(X) at some/any prime above the primeX =x of A=R[X]. The conclusion of Theorem 2.6 holdsa fortiori withLx = (bkF)x. Thus in Theorem 4.1 one can take

L=bkF.

Furthermore,L=k(X) and soL_x =k, under each of the two assumptions (a-1) and (a-2) from Addendum 2.8. We rewrite them in the family context:

(a-1) F is irreducible in k(X)(T)[Y] and the splitting field of F over k(X, T) is regular over k(X),

(a-2) G =Sn.

In these cases, the condition from Theorem 4.1 thatp_i be totally split in the extensionLx/Qreduces to pi being totally split ink/Q,i= 1, . . . , N.

5. Proof of Theorem 2.6 and Addendum 2.8. We return to the general hypotheses of§2.3:Ais a Dedekind domain with fraction fieldk,Pin A[T, Y] is a polynomial, monic and separable inY, and n= deg_Y(P)≥1.

We freely use the notation introduced in§2.

Assume kis of characteristic 0 or greater than deg(∆_P). Letp⊂Abe a good prime ofP (i.e. B_P =∆P,0·disc(∆^red_P )∈/p) and such that|G|∈/p.

5.1. Good Behaviour part. Consider the irreducible factorization

(∗∗) ∆_P =δ

`

Y

h=1

Q^α_h^h

of ∆P ∈ A[T] in the unique factorization domain Ap[T]; δ ∈ Ap \ {0}, Q₁, . . . ,Q<sub>`</sub> are the distinct irreducible factors in Ap[T]\Ap and α1, . . . , α`

are positive integers. As ∆P,0 is invertible in Ap, so are δ and the leading coefficients of Q₁, . . . ,Q_{`</sub>. Hence one can take δ = ∆P,0 and assume that Q<sub>1</sub>, . . . ,Q<sub>`} are monic. We deduce that

∆^red_P = (∆P,0)^ρ

`

Y

h=1

Q_h

whereρ is the number of distinct roots of∆P ink. Mod out (∗∗) byp, and note thats_p(∆_P) =∆_sp(P) and thats_p(∆_P,0) is the leading term∆_sp(P),0 of

∆_sp(P) to conclude that∆_sp(P)6= 0 and

∆_sp(P)=∆_sp(P),0

`

Y

h=1

sp(Q_h)^αh.

For h = 1, . . . , `, each polynomial sp(Q<sub>h</sub>) has only simple roots in κp, and for distinct h, h<sup>0</sup> ∈ {1, . . . , `}, we have s_p(Q_h)6=s_p(Q_h⁰): indeed, otherwise disc(s_p(∆^red_P )) =s_p(disc(∆^red_P )) would be 0, contradicting disc(∆^red_P )∈/ p. It follows thatsp(P) satisfies condition (∗) from Lemma 2.1 and

∆^red_sp(P)= (∆_sp(P),0)^ρ

`

Y

h=1

sp(Q_h) =sp(∆^red_P ).

We conclude that disc(∆^red_s

p(P)) =s_p(disc(∆^red_P )) andB_sp(P)=s_p(B_P).

5.2. The Grothendieck good reduction theorem. We recall below the good reduction criterion for covers, due to Grothendieck, in the context where we will be using it: finite extensions ofk(T). For consistency we stick to the field extension terminology but indicate in footnotes the original geometric formulation.

Assume from now on thatP is irreducible ink(T)[Y]. Denote byF/k(T) the function field extension associated with the polynomialP ∈k(T)[Y] (⁸).

It is regular over k. Let t1, . . . , tr ∈ P¹(k) be the branch points of F/k(T) (more exactly of F k/k(T)). Recall that t₁, . . . , t_r are elements of the set {t₁, . . . , t_ρ,∞}witht₁, . . . , t_ρ the distinct roots of∆_P (but not all elements of this set are branch points in general).

Given a prime idealp⊂A, denote the completion ofA(resp. ofk) atp by Ae_p (resp. byek_p), the algebraic closure ofek_p byC_p and fix an embedding k⊂C_p.

Fix a prime idealp⊂Aand let B be the integral closure ofAep[T] in the field Fekp.

(⁸) Geometrically,F/k(T) corresponds to a branched coverf:C →P¹k.

Grothendieck good reduction theorem. Assume that:

(a) |G|∈/p,

(b) there is no vertical ramification at p in the extension F/k(T), i.e.

B is unramified over the prime pAep (⁹),

(c) no two different branch points of F/k(T) coincide modulo p.

Then the extension F/k(T) has good reduction at p, that is, pB is a prime ideal ofB and the fraction fieldεofB/pB is a separable extension ofκ_p(T) and satisfies

[ε:κ_p(T)] = [κ_pε:κ_p(T)] = [F :k(T)] = deg_Y(P) (¹⁰).

This is a special case of the general result of Grothendieck on reduc- tion of covers. We refer to [Gro71] and [GM71]. In our proof, we apply Grothendieck’s theorem to both the extension F/k(T) and its Galois clo- sure. To this end, the following lemma is useful.

Lemma5.1. Under the assumption|G|B_P ∈/ p, both the extensionF/k(T) and its Galois closureF /k(Tb ) satisfy conditions(a)–(c)of the Grothendieck good reduction theorem.

Proof. Assume |G|B_P ∈/ p. Consider the extension F/k(T). Grothen- dieck’s assumption (a) obviously holds. Assumption (b) is guaranteed by s_p(∆_P) 6= 0 in κ_p[T] (which follows from s_p(∆_P,0) 6= 0). Assumptions (a) and (c) then automatically hold forF /k(Tb ), which has the same monodromy group and the same branch points asF/k(T). It remains to show assumption (b) forF /k(Tb ).

LetY₁,Y₂ be two distinct roots ofP ink(T). The discriminant∆_P is in A[T] and is the discriminant of 1,Y₁, . . . ,Y₁ⁿ⁻¹. LetB₁be the integral closure ofA[T] ink(T,Y₁), andP2∈k(T,Y₁)[Y] be the irreducible polynomial ofY₂ overk(T,Y₁). The discriminant∆P2 ofP2is inB1 and is the discriminant of 1,Y₂, . . . ,Y₂ⁿ²⁻¹forn₂= [k(T,Y₁,Y₂) :k(T,Y₁)]. The polynomialP₂divides P in B₁[Y]. Consequently, ∆_P2 divides ∆_P in B₁, and its norm N₁(∆_P2) relative to the extension k(T,Y₁)/k(T) divides ∆ⁿ_P in A[T]. Consider the set {Y₁^uY₂^v | 0 ≤ u < n, 0 ≤ v < n2}. It is a k(T)-basis of k(T,Y₁,Y₂) and its discriminant ∆₂, in A[T], is a product of some power of ∆_P and some power ofN₁(∆_P2). Therefore, since the leading term of∆_P is not inp, neither is the leading term of ∆2. Repeat the argument on the extensions obtained by adjoining successively every root ofP ink(T), and so eventually on the Galois extensionF /k(T). We finally obtain this conclusion:b

(⁹) Geometrically, the normalization ˜f:C →e P¹_Aep ofP¹_Aep inF is unramified overp.

(¹⁰) Geometrically, if ˜f0 : Ce0 →P¹κ_p is the special fiber of ˜f, then ˜f0 is generically

´etale,Ce0 is geometrically irreducible and [κp(Ce0) :κp(T)] = [F :k(T)].

(∗) There exists ak(T)-basis of the extensionF /k(Tb )consisting of prod- ucts of powers of the roots Y₁, . . . ,Y_n of P in k(T) such that the discriminant, a polynomial ∆b in A[T], has leading term not in the idealp.

This guarantees that there is no vertical ramification at p in the extension F /k(T).b

5.3. Proof of the GRT part of Theorem 2.6. That sp(P) is monic inY is obvious and the separability follows from∆_sp(P)6= 0. For the rest of the proof, denote:

• by Y₁, . . . ,Y_n the roots of P ink(T), and assume thatY₁ ∈F,

• by Ae^bk_p^Pekp the integral closure ofAe_p inbk_Pek_p,

• by Bb the integral closure ofAe^bk_p^Pekp[T] inFbekp,

• by (bk_P)_p/κ_pthe residue extension ofbk_P/kat some/any prime abovep.

The reduction morphism

Ae^bk_p^Pekp[T]→(bk_P)_p(T)

that extends the reduction mapAe^bk_p^Pekp →(bk_P)_p and sendsT to itself can be extended to a morphism

sp :Bb→κp(T) (e.g. [D`eb09, lemme 1.7.2]).

From Grothendieck’s theorem, pB is a prime ideal of B and the fraction field εof B/pB is a separable extension of κp(T). As P is monic, Y₁ ∈ B and one can further assume that s_p(Y₁) is the image ofY₁ in

B/pB ⊂Frac(B/pB)⊂κ_p(T)

(s_p(Y₁) can be chosen to be any root of s_p(P): see e.g. [D`eb09, §1.7.3]).

Hence sp(Y₁)∈ε. The ringAep[T] being integrally closed, we also have

∆_PB ⊂Ae_p[T] +Ae_p[T]Y₁+· · ·+Ae_p[T]Y₁ⁿ⁻¹, which, together withs_p(∆_P)6= 0 inκ_p[T], leads to

ε= Frac(κ_p[T, s_p(Y₁)]).

Ass_p(Y₁) is a root of s_p(P) and from Grothendieck’s theorem, [κ_pε:κ_p(T)] = deg_Y(P) = deg_Y(s_p(P)),

we conclude that the polynomial s_p(P) is irreducible in κ_p[T, Y].

In a similar manner, using conclusion (∗) from the proof of Lemma 5.1, we obtain

Frac(sp(B)) = Frac (b bkP)p[T, sp(Y₁), . . . , sp(Y_n)]

.

From Grothendieck’s theorem applied to the extension F /b bkP(T) (which is regular overbkP),pBb is a prime ideal of B, the fraction fieldb bεofB/pb Bb is a separable extension of (bk_P)_p(T), and

[εb: (bk_P)p(T)] = [κpbε:κp(T)] = [F kb :k(T)].

The extension F /bb k_P(T) being Galois, the residue field extension above p does not depend on the prime ideal abovep. Hence Frac(s_p(Bb)) =ε.b

Consequently, bεis the splitting field of the polynomials_p(P) over (bk_P)_p. The Galois group Gal(κ_pε/κb _p(T)) is by definition the monodromy group of sp(P). It is a prioria subgroup ofG, but is in fact all ofG, since its order is

[κ_pεb:κ_p(T)] = [F kb :k(T)] =|G|.

5.4. Proof of Addendum 2.8. (a) Item (a-1) follows from the defini- tions, and (a-2) from the fact that ifG=S_n, then necessarily G_a isS_n too.

(b) From the equality

[bε: (bkP)p(T)] = [κpεb:κp(T)]

shown above, the extension ε/(bb k_P)_p(T) is regular over (bk_P)_p. This implies that (bkP)p contains the constant extension in ε/κb p(T) (i.e. in the splitting field ofsp(P) over κp(T)).

(c) Assume k is a number field. If p is a good prime of P, then the ex- tension F /k(Tb ) has good reduction at p. By a result of Deligne–Mumford [DM69, Theorem 1.3], the k-automorphisms of the extension F k/k(Tb ) are already defined over the unramified closure ek^ur_p of ek_p. Hence the extension Febk^ur_p /ek^ur_p (T) is Galois, which implies that bk_P ⊂ ek^ur_p . This shows that the ramified primes in bk_P/k are among the bad primes. Therefore bk_P/k is one from the finite list of extensions ofkof degree ≤ |Nor_Sn(G)|/|G|and unram- ified at each good prime ofP.

5.5. Proof of the Chebotarev part of Theorem 2.6. Assume p is good, not containing |G|, of norm N_k/Q(p) ≥ (ρ+ 1)²|G|² and totally split in the extension bk_P/k. Let ω ∈ G and letE_ω/ek_p be the unique unramified extension of Galois grouphωi ⊂ G.

The result follows from [DG12, Proposition 2.2] applied to the Galois extensionFbekp/ekp(T) (viewed there as a G-cover) and the Galois extension E_ω/ek_p. The base ring A there should be taken to be the ring Ae^bk_p^Pekp intro- duced in §5.3. The situation in [DG12] is that of a cover of a more general base space, of arbitrary dimension. We review the proof below, which be- comes simpler for a cover ofP¹.